Copied to
clipboard

## G = (C2×C42).C4order 128 = 27

### 6th non-split extension by C2×C42 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C42).C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.67C23 — (C2×C42).C4
 Lower central C1 — C2 — C22 — C2×C4 — (C2×C42).C4
 Upper central C1 — C22 — C23 — C2×C4⋊C4 — (C2×C42).C4
 Jennings C1 — C22 — C23 — C2×C4⋊C4 — (C2×C42).C4

Generators and relations for (C2×C42).C4
G = < a,b,c,d | a2=b4=c4=1, d4=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab-1c-1, dcd-1=ab2c-1 >

Subgroups: 168 in 67 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C22⋊C8, C2×C42, C2×C4⋊C4, C22×Q8, C22.M4(2), C23.67C23, (C2×C42).C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, C423C4, C42.3C4, (C2×C42).C4

Character table of (C2×C42).C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 -i i -i -i -i i i i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 i -i i i i -i -i -i linear of order 4 ρ7 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 -i -i -i i i i i -i linear of order 4 ρ8 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 i i i -i -i -i -i i linear of order 4 ρ9 1 -1 1 -1 1 -1 i i -i -1 -i i -i -i i 1 1 -1 ζ87 ζ8 ζ83 ζ87 ζ83 ζ85 ζ8 ζ85 linear of order 8 ρ10 1 -1 1 -1 1 -1 i i -i -1 -i i -i -i i 1 1 -1 ζ83 ζ85 ζ87 ζ83 ζ87 ζ8 ζ85 ζ8 linear of order 8 ρ11 1 -1 1 -1 1 -1 i -i i -1 i i -i -i -i 1 -1 1 ζ8 ζ83 ζ85 ζ85 ζ8 ζ83 ζ87 ζ87 linear of order 8 ρ12 1 -1 1 -1 1 -1 i -i i -1 i i -i -i -i 1 -1 1 ζ85 ζ87 ζ8 ζ8 ζ85 ζ87 ζ83 ζ83 linear of order 8 ρ13 1 -1 1 -1 1 -1 -i i -i -1 -i -i i i i 1 -1 1 ζ87 ζ85 ζ83 ζ83 ζ87 ζ85 ζ8 ζ8 linear of order 8 ρ14 1 -1 1 -1 1 -1 -i -i i -1 i -i i i -i 1 1 -1 ζ8 ζ87 ζ85 ζ8 ζ85 ζ83 ζ87 ζ83 linear of order 8 ρ15 1 -1 1 -1 1 -1 -i i -i -1 -i -i i i i 1 -1 1 ζ83 ζ8 ζ87 ζ87 ζ83 ζ8 ζ85 ζ85 linear of order 8 ρ16 1 -1 1 -1 1 -1 -i -i i -1 i -i i i -i 1 1 -1 ζ85 ζ83 ζ8 ζ85 ζ8 ζ87 ζ83 ζ87 linear of order 8 ρ17 2 2 2 2 2 2 0 2 -2 -2 2 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 2 0 -2 2 -2 -2 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 0 -2i -2i 2 2i 0 0 0 2i -2 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 2 -2 2 -2 0 2i 2i 2 -2i 0 0 0 -2i -2 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 4 -4 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 4 -4 -4 0 0 -2 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ24 4 4 -4 -4 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ25 4 -4 -4 4 0 0 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ26 4 -4 -4 4 0 0 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4

Smallest permutation representation of (C2×C42).C4
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(2 19 32 12)(3 29)(4 17 26 10)(6 23 28 16)(7 25)(8 21 30 14)(9 20)(13 24)
(1 15 31 22)(2 19 32 12)(3 24 25 9)(4 14 26 21)(5 11 27 18)(6 23 28 16)(7 20 29 13)(8 10 30 17)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,19,32,12)(3,29)(4,17,26,10)(6,23,28,16)(7,25)(8,21,30,14)(9,20)(13,24), (1,15,31,22)(2,19,32,12)(3,24,25,9)(4,14,26,21)(5,11,27,18)(6,23,28,16)(7,20,29,13)(8,10,30,17), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,19,32,12)(3,29)(4,17,26,10)(6,23,28,16)(7,25)(8,21,30,14)(9,20)(13,24), (1,15,31,22)(2,19,32,12)(3,24,25,9)(4,14,26,21)(5,11,27,18)(6,23,28,16)(7,20,29,13)(8,10,30,17), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32) );

G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(2,19,32,12),(3,29),(4,17,26,10),(6,23,28,16),(7,25),(8,21,30,14),(9,20),(13,24)], [(1,15,31,22),(2,19,32,12),(3,24,25,9),(4,14,26,21),(5,11,27,18),(6,23,28,16),(7,20,29,13),(8,10,30,17)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)]])

Matrix representation of (C2×C42).C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 16 0 16 7 0 0 14 0 7 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 8 14 0 0 0 0 16 9 0 0 0 0 2 6 16 7 0 0 11 11 7 1
,
 0 9 0 0 0 0 8 0 0 0 0 0 0 0 13 0 15 0 0 0 0 0 1 1 0 0 0 1 4 0 0 0 1 0 13 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,16,14,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,7,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,8,16,2,11,0,0,14,9,6,11,0,0,0,0,16,7,0,0,0,0,7,1],[0,8,0,0,0,0,9,0,0,0,0,0,0,0,13,0,0,1,0,0,0,0,1,0,0,0,15,1,4,13,0,0,0,1,0,0] >;

(C2×C42).C4 in GAP, Magma, Sage, TeX

(C_2\times C_4^2).C_4
% in TeX

G:=Group("(C2xC4^2).C4");
// GroupNames label

G:=SmallGroup(128,51);
// by ID

G=gap.SmallGroup(128,51);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,723,352,1242,521,136,2804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b^-1*c^-1,d*c*d^-1=a*b^2*c^-1>;
// generators/relations

Export

׿
×
𝔽